Environmental Engineering Reference
In-Depth Information
where the threshold frequency of photodetachment is
„
ω
D
EA
D
0.754 eV, and
10
17
cm
2
and is attained at the frequen-
the maximum cross section is
σ
D
4
max
cy
ω
D
2
ω
0
, which corresponds to the photon energy
„
ω
D
2EA
D
1.51 eV
max
max
or to the photon wavelength
m.
Using parameters of the average quiet solar photosphere [66, 67], we have
the effective temperature of radiation
T
ef
λ
D
0.8
μ
5780 K if we model the Sun's sur-
face by a blackbody, and the free-fall acceleration on the solar boundary is
g
D
D
10
4
cm
2
/s. This gives the length
l
2.74
D
174 km for the barometric distribu-
10
21
cm
3
tion (1.48) of hydrogen atoms and
N
0
D
1.06
in (3.96). According
to the definition, the boundary of the photosphere,
z
D
0, is given as a layer
with optical thickness
u
(
λ
D
0.5
μ
m)
D
1. At this wavelength we have the fac-
tor exp(
0.008 in (2.148). Ignoring this factor and taking according
to the barometric formula (1.48)
N
H
(
z
)
„
ω
max
/
T
)
D
exp(
z
/
l
) and according to (3.96)
N
(
z
)
D
N
(0) exp(
3
z
/2
l
), we have the optical density of the photosphere
2
l
3
N
(
z
)
u
ω
(
z
)
D
σ
det
(
ω
) ,
(3.100)
where
z
is the photosphere altitude. From the condition
u
(
z
D
0,
λ
D
0.5
μ
m)
D
1
and accounting for
max
, we obtain for the number density
of negative hydrogen ions at the photosphere boundary
N
(0)
σ
det
(
λ
D
0.5
μ
m)
D
0.8
σ
10
9
cm
3
.
Then (3.96) gives for the number densities of hydrogen atoms and electrons at the
photosphere boundary
N
H
(0)
D
2.7
10
17
cm
3
and
N
e
(0)
10
13
cm
3
.
D
1.6
D
1.5
2/3 gives the altitude
z
ω
of the
layer that is responsible for radiation at frequency
In accordance with (3.97), the condition
u
ω
D
ω
. On the basis of (3.100) for
the optical thickness we obtain
z
max
D
73 km is responsible for radiation at
λ
D
0.8
m, which corresponds to the maximum photodetachment cross section. At
this altitude we have
N
(
z
max
)
μ
10
9
cm
3
,
N
H
(
z
max
)
10
17
cm
3
,
D
1.4
D
1.1
10
13
cm
3
. Next, the photosphere altitude responsible for
and
N
e
(
z
max
)
D
1.2
emission at
23 km.
In these cases the photon flux at a given frequency is determined by (1.61). In
addition to this, we have that radiation of the solar photosphere at frequencies of
the order of the maximum frequency is determined by the photosphere layer of
width
λ
D
0.5
μ
mis
z
ω
D
47 km, and at
λ
D
0.4
μ
mitis
z
ω
D
Δ
z
1/
k
ω
200km.Hence,onecanignorethetemperaturevariationin
the radiating layer if
T
2
dT
dz
„
ω
10 K/km .
Δ
z
If the photosphere temperature varies slightly with the altitude, we use in (3.97)
for the photon flux the temperature at altitude
z
ω
. Summing the radiative flux of
the Sun's photosphere
j
ω
from radiation of layers with a different temperature, we
use (3.97) as an expansion of
j
ω
over a small parameter, and below we find the
validity of this expansion where the second term in parentheses in (3.97) is much
less than 1. We have
N
(
z
)
exp(
3
z
/2
l
ε
/
T
), which gives
1
du
dz
D
3
u
2
l
dT
dz
2
l
3
ε
0
T
2
α
,
α
D
,
